What are wrong with this phrase?

Solution 1:

"What is wrong with XYZ?" is valid regardless of whether XYZ is one thing or many, and whether the questioner expects an answer detailing one fault or many.

If the questioner wanted to explicitly indicate that he expects an answer listing multiple faults, he'd have to say something like "What things are wrong with XYZ?".

"What are wrong with XYZ?" is never grammatically valid. Nor is it a common mistake — I don't recall seeing it before now, and Google Books records only 14 instances of "what are wrong with", compared to millions of "what is wrong with".

As regards exactly why the non-standard usage is unacceptable, rather than just uncommon, what in this construction is a non-count pronoun. Non-count nouns require the singular verb form.

Edit: Kudos to JLG for highlighting the importance of the word wrong in this construction. The interrogative pronoun what attaches to wrong — a non-count abstract noun which transfers its non-count status to what. That doesn't happen with, for example, "What are those things?".

Solution 2:

"What" can be either singular or plural. "is/are" is a linking verb, so the number of the things on either side -- the subject and the predicate nominative -- should match, and then this number should match the verb.

So, for example, we say, "What IS the name of your friend?" because "name" is singular, so "what" is being used as a singular, so the verb should be singular. But we say, "What ARE the names of your friends?" because "names" is plural, etc.

In this case we have a predicate adjective rather than a predicate nominative. I think in such cases the subject "what" is always considered singular. I'm trying to think of exceptions. So in "What is/are wrong with X, Y, and Z", we should use "is". I guess you are being thrown off by the prepositional phrase "with X, Y, and Z". But such a phrase does not affect the number of the subject. A simple rule of thumb is, when trying to determine things about the subject and verb, just ignore any "extra detail" phrases.

If that doesn't make sense to your intuition, consider this sentence, "Who is/are the policeman who arrested Smith and Jones?" Clearly it should be "is", because "policeman" is singular -- we are asking for one policeman. The fact that "Smith and Jones" are two people is irrelevant. They're not the "who" here: "the policeman" is.

In the same way, the thing you are asking about in your sentence is "wrong". The fact that it is "with X, Y, and Z" has nothing to do with the number of the subject.